Yes, that green wire normally feeds into a ballast resistor and then from ballast to the + side of the coil. Since it looks like you are using a blue coil that doesn't require the ballast resistor, it should connect directly to the + side of the coil. That wire gets bypassed when starting, that's why the car starts but then dies when you're not in starting mode. Make sure that the wire is getting 12+volts when the key is turned ON, but not turned to START.
BTW, that box that the green wire is connected to is called a relay...